Normalizers of maximal tori and classifying spaces of compact Lie groups (Q2739719)

The author extends a result by \textit{A. Osse} [C. R. Acad. Sci., Paris. Sér. I 315, No. 7, 833-838 (1992; Zbl 0779.55011)] to the case of non-connected Lie groups. He showsNEWLINENEWLINENEWLINETheorem. Two (not necessarily connected) compact Lie groups are isomorphic if and only if their classifying spaces are homotopy equivalent.NEWLINENEWLINENEWLINEOn the way to proving this last theorem, he also gets the following result for the normalizer \(N_0\):NEWLINENEWLINENEWLINETheorem. The map \(\beta_{N_0}: \text{Out}(N_0) \to\Aut(BN_0)\), \([\psi]\mapsto [B\psi]\) is an isomorphism of groups. In particular \(\Aut(BN_0) \cong H^1(W_0;T) \rtimes\text{Out}(G_0)\). NEWLINENEWLINENEWLINEIn other words, the group of homotopy classes of self-homotopy equivalences of \(BN_0\) and the outer automorphism group of \(N_0\) are isomorphic. This extends to the class of normalizers \(N_0\) a result for connected compact Lie groups of \textit{S. Jackowski}, \textit{J. McClure} and \textit{B. Oliver} [Fundam. Math. 147, No. 2, 99-126 (1995; Zbl 0835.55012)].